Newform orbit 29.6.d.a

• Label: 29.6.d.a
• Level: $29$
• Weight: $6$
• Character orbit: 29.d
• Analytic conductor: $4.651$
• Analytic rank: $0$
• Dimension: $66$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	29.d (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.65113077458\)
Analytic rank:	\(0\)
Dimension:	\(66\)
Relative dimension:	\(11\) over \(\Q(\zeta_{7})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{7}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 11 q^{2} - 5 q^{3} - 171 q^{4} + 29 q^{5} + 445 q^{6} + 17 q^{7} + 1046 q^{8} - 732 q^{9}+O(q^{10}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−2.41509	−	10.5812i	0.690047	+	0.332309i	−77.2985	+	37.2250i	19.1233	+	83.7844i	1.84971	−	8.10409i	−158.450	−	76.3053i	364.027	+	456.475i	−151.142	−	189.526i	840.357	−	404.695i
7.2	−1.89325	−	8.29487i	14.6426	+	7.05150i	−36.3895	+	17.5242i	−19.7959	−	86.7313i	30.7692	−	134.809i	23.9398	+	11.5288i	44.5031	+	55.8051i	13.1736	+	16.5192i	−681.946	+	328.408i
7.3	−1.73099	−	7.58398i	−20.8543	−	10.0429i	−25.6894	+	12.3713i	−4.14155	−	18.1453i	−40.0664	+	175.543i	89.9242	+	43.3052i	−16.9122	−	21.2072i	182.533	+	228.889i	−130.445	+	62.8189i
7.4	−1.10867	−	4.85738i	13.3571	+	6.43244i	6.46597	−	3.11385i	14.7871	+	64.7867i	16.4363	−	72.0120i	140.749	+	67.7811i	−121.699	−	152.606i	−14.4722	−	18.1476i	298.300	−	143.654i
7.5	−0.594851	−	2.60621i	−6.88516	−	3.31572i	22.3925	−	10.7837i	−1.88513	−	8.25929i	−4.54582	+	19.9165i	−133.434	−	64.2584i	−94.7603	−	118.826i	−115.097	−	144.327i	−20.4041	+	9.82609i
7.6	0.402095	+	1.76169i	22.1010	+	10.6433i	25.8891	−	12.4675i	−5.52960	−	24.2268i	−9.86349	+	43.2148i	−77.8191	−	37.4757i	68.4265	+	85.8041i	223.667	+	280.469i	40.4567	−	19.4829i
7.7	0.522777	+	2.29044i	−22.0166	−	10.6026i	23.8582	−	11.4895i	18.8051	+	82.3907i	12.7749	−	55.9704i	123.108	+	59.2858i	85.6617	+	107.416i	220.807	+	276.883i	−178.880	+	86.1439i
7.8	0.813916	+	3.56600i	−6.23245	−	3.00139i	16.7771	−	8.07943i	−22.0176	−	96.4653i	5.63027	−	24.6678i	166.222	+	80.0483i	115.444	+	144.762i	−121.673	−	152.573i	326.075	−	157.029i
7.9	1.27150	+	5.57079i	5.91325	+	2.84767i	−0.585948	+	0.282178i	13.3453	+	58.4697i	−8.34510	+	36.5623i	−1.93045	−	0.929655i	111.688	+	140.052i	−124.651	−	156.307i	−308.754	+	148.688i
7.10	1.95082	+	8.54710i	−16.4262	−	7.91043i	−40.4162	+	19.4634i	−4.08126	−	17.8812i	35.5667	−	155.828i	−139.508	−	67.1835i	−70.2862	−	88.1361i	55.7366	+	69.8915i	144.870	−	69.7659i
7.11	2.19360	+	9.61081i	14.0872	+	6.78404i	−58.7248	+	28.2804i	0.197313	+	0.864484i	−34.2984	+	150.271i	69.1061	+	33.2798i	−203.933	−	255.724i	0.918137	+	1.15131i	−7.87556	+	3.79267i
16.1	−9.29727	−	4.47733i	−16.9028	+	21.1954i	46.4410	+	58.2352i	−29.3057	−	14.1129i	252.049	−	121.380i	−125.979	+	157.972i	−97.5569	−	427.425i	−109.469	−	479.616i	209.275	+	262.422i
16.2	−9.01987	−	4.34374i	7.86720	−	9.86516i	42.5383	+	53.3413i	61.9634	+	29.8400i	−113.813	+	54.8094i	56.6689	−	71.0605i	−80.7019	−	353.578i	18.6441	+	81.6852i	−429.284	−	538.306i
16.3	−6.13336	−	2.95367i	1.76173	−	2.20914i	8.94225	+	11.2132i	−38.8583	−	18.7131i	−17.3304	+	8.34587i	0.981087	−	1.23024i	26.7482	+	117.192i	52.2960	+	229.124i	183.059	+	229.549i
16.4	−3.39646	−	1.63565i	−13.4299	+	16.8406i	−11.0911	−	13.9078i	21.4922	+	10.3501i	73.1594	−	35.2317i	143.536	−	179.989i	41.7656	+	182.987i	−49.1696	−	215.426i	−56.0684	−	70.3076i
16.5	−3.07802	−	1.48230i	18.8759	−	23.6696i	−12.6747	−	15.8935i	15.8039	+	7.61077i	−93.1857	+	44.8759i	−39.8314	+	49.9470i	39.7806	+	174.290i	−149.879	−	656.661i	−37.3634	−	46.8523i
16.6	−1.37713	−	0.663191i	−4.45689	+	5.58876i	−18.4950	−	23.1920i	71.8910	+	34.6209i	9.84412	−	4.74068i	−153.020	+	191.881i	20.9732	+	91.8897i	42.7022	+	187.091i	−76.0429	−	95.3548i
16.7	1.69866	+	0.818031i	5.27812	−	6.61856i	−17.7354	−	22.2395i	−69.8948	−	33.6596i	14.3799	−	6.92500i	10.7481	−	13.4777i	−25.3589	−	111.105i	38.1259	+	167.040i	−91.1929	−	114.352i
16.8	3.74798	+	1.80493i	−13.9558	+	17.5000i	−9.16211	−	11.4889i	−40.2504	−	19.3836i	−83.8924	+	40.4005i	−30.7029	+	38.5002i	−43.2242	−	189.378i	−57.4139	−	251.547i	−115.872	−	145.298i
16.9	3.98347	+	1.91834i	6.52648	−	8.18394i	−7.76363	−	9.73529i	64.0881	+	30.8632i	41.6976	−	20.0805i	85.0547	−	106.655i	−43.7334	−	191.609i	29.6906	+	130.083i	196.087	+	245.886i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
29.d	even	7	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	29.6.d.a	✓	66
29.d	even	7	1	inner	29.6.d.a	✓	66
29.d	even	7	1		841.6.a.i		33
29.e	even	14	1		841.6.a.h		33

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
29.6.d.a	✓	66	1.a	even	1	1	trivial
29.6.d.a	✓	66	29.d	even	7	1	inner
841.6.a.h		33	29.e	even	14	1
841.6.a.i		33	29.d	even	7	1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(29, [\chi])\).